Finite Group Databasebeta

← #30650650

Subgroups — #30650650

Order 16256.

386 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550599C211yesmaximal, characteristic, complemented
2#22550529C211yesmaximal, characteristic
3#22550226C211yesmaximal, characteristic, complemented
4#22550216C212yesmaximal, complemented
5#22550157C211yesmaximal, characteristic, complemented
6#22550114C212yesmaximal, complemented
7#22550037C212yesmaximal, complemented
8C4×#14523235C211yesmaximal, characteristic, complemented
9#22549941C211yesmaximal, characteristic, complemented
10#22549871C212yesmaximal, complemented
11#462×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2212yes—
13C2×#14523235C2211yescharacteristic
14C2×C8×D127C2212yes—
15#16861209C2211yescharacteristic
16#16861204—22no—
17#16861197—21no—
18#16861194—22no—
19#16861185C2212yes—
20#16861185C2212yes—
21#16861184—21no—
22C4×Dic254—21no—
23C4×D508C2211yescharacteristic
24C4×D508C2211yescharacteristic
25#16861167C2212yes—
26C4⋊C4×D127—21no—
27#16861143C2212yes—
28#16861140C2211yescharacteristic
29#16861133C2212yes—
30#16861133C2211yescharacteristic
31#16861133C2212yes—
32#16861131C2212yes—
33#16861129C2211yescharacteristic
34(C2×D127)⋊C8C2212yes—
35C4×C127⋊C8C2211yescharacteristic
36C2×Q16×C127C2212yes—
37C4⋊Q8×C127C2211yescharacteristic
38C4×Q8×C127C2211yescharacteristic
39C4⋊C8×C127C2211yescharacteristic
40#103×C127C2212yes—
41#14523249—24no—
42#14523249—42no—
43C2×Dic254—21no—
44C2×D508—22no—
45C2×D508D411yescharacteristic
46C2×D508C2311yescharacteristic
47C2×C4×D127—22no—
48C2×C4×D127—21no—
49C2×C4×D127C2312yes—
50#14523235—22no—
51C2×C127⋊C8C2311yescharacteristic
52C2×C127⋊C8C2311yescharacteristic
53C8×D127—42no—
54C4⋊Dic127—21no—
55C4⋊Dic127D411yescharacteristic
56C4⋊Dic127C2311yescharacteristic
57C4⋊Dic127C2312yes—
58Dic127⋊C4—22no—
59Dic127⋊C4—22no—
60(C2×D127)⋊C4—22no—
61(C2×D127)⋊C4—22no—
62(C2×D127)⋊C4—22no—
63(C2×D127)⋊C4—21no—
64(C2×D127)⋊C4D412yes—
65C4×Dic127—22no—
66C2×Q8×C127C2311yescharacteristic
67C2×Q8×C127C2312yes—
68Q16×C127—42no—
69C2×C8×C127C2312yes—
70C4⋊C4×C127—21no—
71C4⋊C4×C127—21no—
72C4⋊C4×C127—21no—
73C4⋊C4×C127C2311yescharacteristic
74C42×C127—21no—
75C42×C127C2311yescharacteristic
76C22×D127—21no—
77C22×D127C2×D412yes—
78Dic254—42no—
79D508—22no—
80D508—21no—
81D508—44no—
82D508#4112yes—
83C2×Dic127—22no—
84C2×Dic127—21no—
85C2×Dic127C2×D412yes—
86C4×D127—22no—
87C4×D127—22no—
88C4×D127—44no—
89C4×D127—42no—
90C127⋊C8—21no—
91C127⋊C8—21no—
92Q8×C127—21no—
93Q8×C127—22no—
94Q8×C127—21no—
95Q8×C127—22no—
96C2×C4×C127—21no—
97C2×C4×C127—21no—
98C2×C4×C127—22no—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2411yescharacteristic
102C8×C127—22no—
103C22×C127C22×D411yescharacteristic
104C2×D127—22no—
105C2×D127—22no—
106C2×D127—22no—
107C2×D127—22no—
108C2×D127—41no—
109C2×D127—41no—
110C4×C127—21no—
111C4×C127—41no—
112C4×C127—42no—
113C4×C127—41no—
114C4×C127#13512yes—
115C4×C127C2×#4111yescharacteristic
116C4×C127H(5,𝔽2)11yescharacteristic
117Dic127—22no—
118Dic127—41no—
119Dic127—42no—
120C2×C127#54611yescharacteristic
121C2×C127C2×#13511yescharacteristic
122C2×C127#57811yescharacteristic
123D127—42no—
124D127—42no—
125#3302—1271nomaximal, Sylow
126C127#330211yescharacteristic, complemented, Sylow
127#556—1271no—
128#547—1271no—
129#487—1271no—
130#485—1272no—
131#474—1271no—
132#467—1272no—
133#462D12711yescharacteristic, complemented
134#452—1272no—
135C4×SD16—1271no—
136C8×D4—1271no—
137#416—1272no—
138C2×#41—1272no—
139C2×Q16C2×D12712yes—
140C2×SD16—1271no—
141C22×C8—1272no—
142C4⋊Q8—1271no—
143C4⋊Q8C2×D12711yescharacteristic
144#126—2542no—
145#124—2541no—
146#123—2542no—
147C22⋊Q8—1272no—
148C22⋊Q8—1272no—
149#121—2541no—
150C4×Q8—2541no—
151C4×Q8C2×D12711yescharacteristic
152C4×D4—1271no—
153C4×D4—1271no—
154#117—1272no—
155C2×C4⋊C4—2541no—
156#107—1272no—
157#106—1271no—
158C4⋊C8C2×D12711yescharacteristic
159#103—1272no—
160#103—1271no—
161#103—1272no—
162#103C2×D12712yes—
163#102—1272no—
164#102—1271no—
165C22⋊C8—1272no—
166C4×C8—1271no—
167#41—5082no—
168#41—2544no—
169C2×Q8—2541no—
170C2×Q8C22×D12712yes—
171C2×Q8C22×D12711yescharacteristic
172C2×D4—1271no—
173C2×D4—1271no—
174C2×D4—2542no—
175C22×C4—1272no—
176C22×C4—2542no—
177C22×C4—2541no—
178Q16—42no—
179SD16—2542no—
180C2×C8—1271no—
181C2×C8—1271no—
182C2×C8—5082no—
183C2×C8C22×D12712yes—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—1271no—
188C4⋊C4—1272no—
189C4⋊C4—1271no—
190C4⋊C4—2542no—
191C4⋊C4—2542no—
192C4⋊C4—2541no—
193C4⋊C4C22×D12711yescharacteristic
194C22⋊C4—1272no—
195C22⋊C4—2542no—
196C22⋊C4—2542no—
197C22⋊C4—2542no—
198C22⋊C4—2541no—
199C42—21no—
200C42—2542no—
201C42C22×D12711yescharacteristic
202C23—1272no—
203C23—2541no—
204Q8—21no—
205Q8—22no—
206Q8—21no—
207Q8—22no—
208Q8—5082no—
209D4—1272no—
210D4—5084no—
211D4—2542no—
212D4—2541no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—1272no—
217C2×C4—5084no—
218C2×C4—5082no—
219C2×C4—2542no—
220C2×C4—2541no—
221C2×C4—2542no—
222C2×C4—2542no—
223C2×C4D4×D12711yescharacteristic
224C2×C4D4×D12711yescharacteristic
225C2×C4C23×D12711yescharacteristic
226C8—22no—
227C8—2541no—
228C8—2541no—
229C22—5081no—
230C22—5081no—
231C22—2542no—
232C22—2542no—
233C22—2542no—
234C22—2542no—
235C22C2×D4×D12711yescharacteristic
236C4—21no—
237C4—41no—
238C4—42no—
239C4—41no—
240C4—5081no—
241C4—5082no—
242C4—2542no—
243C4#1686123212yes—
244C4#41×D12711yescharacteristic
245C4#1686125411yescharacteristic
246C2—5082no—
247C2—5082no—
248C2#2255052311yescharacteristic
249C2C2×#1686123211yescharacteristic
250C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#462×C127C2
socleC22×C127C22×D4
prime radical#22550529C2
hypercenter#462D127
hypocenterC127#3302
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22550599C2semidirect
3. #22550226C2semidirect
4. #22550216C2semidirect
5. #22550157C2semidirect
6. #22550114C2semidirect
7. #22550037C2semidirect
8. C4×#14523235C2semidirect
9. #22549941C2semidirect
10. #22549871C2semidirect
11. #462×C127C2semidirect
126. C127#3302semidirect
133. #462D127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225505992
#225505292
#225502262
#225502162
#225501572
#225501142
#225500372
C4×#145232352
#225499412
#225498712
#462×C1272
#3302127